Section: Scientific Foundations
Challenges related to direct numerical simulations of turbulent flows
In addition to the classical challenges of DNS in very simple situations, the extension to more involved geometries and boundary conditions raises new questions in this field.

Size of discrete problems The resolution required to attain the smallest scales of a turbulent flow field implies a tremendous number of mesh points and time steps, which has restricted DNS simulations for a long time to extremely simple geometries.

Solution of discrete problems The minimum amount of work to be done in each step of time iteration is the solution of the discrete pressure equation. Due to the mesh resolution, a fast solver with linear complexity is required for this elliptic secondorder equation. Our approach is based on nonconforming finite elements which lead to a favorable structure of the pressure equation.

General hexahedral meshes Standard staggered finite difference schemes, which are closely related on cartesian meshes to our approach, may lead to loss of accuracy on distorted hexahedral meshes. However, such meshes are unavoidable for more involved geometries.

Anisotropic meshes In order to resolve boundary layers, the use of anisotropic meshes is mandatory. This leads to the question of stability of the discretization on such meshes. The solution of the pressure equation requires additional care compared to the isotropic case.

Higher accuracy Higherorder schemes have been shown to be successful in certain situations. We intend to generalize our secondorder scheme to higherorder, including general hexahedra and curved boundaries.
Recently, it has been observed that certain turbulence models have similarities to finite element stabilization techniques for the NavierStokes equations. Such variational multiscale methods lead to adaptive turbulence modeling, which are however far from being a standard tool for turbulent flow simulations nowadays. We intend to contribute to this development, and the development of our DNS code is an important ingredient.